??? Associativity for any binary operation is just (a*b)*c= a*(b*c). The point is that this allows us to extend the binary operation (defined for two operands) unambiguously to any number. I am not sure what you mean by "explain it for multiplication modulo n".

Perhaps this: let a, b c be numbers. Then (ab)c= a(bc) (mod n) if and only if (ab)c- a(bc) (mod n) which is the same as saying that (ab)c- a(bc), as an "ordinary" number, is a multiple of n. But in fact, because the usual multiplication of integers is associative, (ab)c= a(bc) so (ab)c- a(bc)= 0= 0(n) is a multiple of n.